### 2.1 Equations

Using the Einstein summation convention the equations describing the motion of a relativistic fluid are
given by the five conservation laws,
where , and where denotes the covariant derivative with respect to coordinate .
Furthermore, is the proper rest mass density of the fluid, its four-velocity, and is the
stress-energy tensor, which for a perfect fluid can be written as
Here, is the metric tensor, the fluid pressure, and the specific enthalpy of the fluid defined by
where is the specific internal energy. Note that we use natural units (i.e., the speed of light c = 1)
throughout this review.
In Minkowski spacetime and Cartesian coordinates , the conservation equations (1, 2) can
be written in vector form as

where i = 1 ,2 ,3. The state vector is defined by
and the flux vectors are given by
The five conserved quantities , , , , and are the rest mass density, the three
components of the momentum density, and the energy density (measured relative to the rest mass energy
density), respectively. They are all measured in the laboratory frame, and are related to quantities in the
local rest frame of the fluid (primitive variables) through
where are the components of the three-velocity of the fluid
and is the Lorentz factor,
The system of Equations (5) with Definitions (6, 7, 8, 9, 10, 11, 12) is closed by means of an equation of
state (EOS), which we shall assume to be given in the form
In the non-relativistic limit (i.e., , ) , , and approach their Newtonian
counterparts , , and , and Equations (5) reduce to the classical ones. In the
relativistic case the equations of system (5) are strongly coupled via the Lorentz factor and the specific
enthalpy, which gives rise to numerical complications (see Section 2.3).

In classical numerical hydrodynamics it is very easy to obtain from the conserved quantities (i.e.,
and ). In the relativistic case, however, the task to recover from is much
more complicated. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the
conserved quantities are advanced in time, it is necessary to compute the primitive variables from the
conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial
ingredient of any algorithm (see Section 9.2).